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“International Portfolio Management”
Ioannis Kalaitzis
Supervisor: Nicolas Topaloglou
Athens University of Economics and Business
Department of International and European Economic
Studies
September 2020, Athens
1
Table of contents 1 Introduction ........................................................................................................................................... 3
2 Kinds of risk .......................................................................................................................................... 4
2.1 Market risk: ................................................................................................................................... 4
2.2 Exchange risk: ............................................................................................................................... 4
2.3 Interest rate risk: ........................................................................................................................... 4
2.4 Credit risk: .................................................................................................................................... 4
2.5 Liquidity risk: ................................................................................................................................ 5
2.6 Inflation risk: ................................................................................................................................. 5
2.7 Settlement risk: ............................................................................................................................. 5
2.8 Sovereign risk: .............................................................................................................................. 5
3 Risk measures and mathematical formulations ..................................................................................... 6
3.1 Variance: ....................................................................................................................................... 8
3.2 Value at Risk (VaR): ..................................................................................................................... 9
3.3 Conditional Value at Risk (CVaR): ............................................................................................ 10
3.4 Mean Absolute Deviation (MAD) .............................................................................................. 12
3.5 Tracking error models ................................................................................................................. 13
3.6 Put/Call Efficient Frontiers Model .............................................................................................. 14
3.7 Expected Utility Maximization Model........................................................................................ 15
4 International Diversification – Does it pay to go internationally? ...................................................... 18
4.1 Empirical Results ........................................................................................................................ 19
5 Empirical Application ......................................................................................................................... 26
5.1 Data Description ......................................................................................................................... 27
5.2 Methodology ............................................................................................................................... 27
5.2.1 Domestic Portfolio .............................................................................................................. 27
5.2.2 International Portfolio ......................................................................................................... 28
5.3 Empirical results ......................................................................................................................... 29
5.3.1 Efficient Frontier ................................................................................................................. 29
5.3.2 Back Testing ....................................................................................................................... 31
5.4 Analysis....................................................................................................................................... 34
5.4.1 Descriptive statistics of returns ........................................................................................... 34
5.4.2 Exchange rates .................................................................................................................... 35
5.5 Home Asset Bias ......................................................................................................................... 35
6 Conclusion .......................................................................................................................................... 37
7 References ........................................................................................................................................... 38
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3
1 Introduction
Risk is something infinitely dimensional and depends on all of the possible events in the world. Financial
advisors, organizations and investors aim to structure their investment strategies combining assets that will
offer a balance between risk and return. Since 1952, when Markowitz published his paper about portfolio
selection in which he tries to maximize portfolio expected return for a given amount of risk, researchers
have created various risk measures for optimal portfolio selection. However, it is not attainable to determine
risk precisely. The only accurate answer we can give to the question “What is the maximal loss we can
suffer over some time horizon?” is, “We can lose everything!” However, the probability of this event is
very small.
The current thesis is separated in a theoretical and an empirical part. As far as the theoretical part concerns,
we analyze the various kinds of risks that an investor faces while investing in financial assets. Secondly,
we cite the risk measures and their mathematical formulation that have been published in literature and are
used commonly in modern economies for the portfolio selection. These measures are applied to financial,
engineering and insurance sector by the managers in order to make optimal decisions. In the last section of
the theoretical part, we investigate the benefits of investing internationally for investors who come from
developed and development countries. The impact of international diversification depends on a variety of
factors that we analyze in that section from a financial and an econometrical point of view, according to the
international literature.
As far as the empirical part concerns, we implement a portfolio optimization problem using CVaR model
as risk measurement. Particularly, we apply the optimization problem on an investor from Germany who
has the option to invest in a domestic and an international portfolio with assets from Germany, U.S.A and
Japan. After receiving the historical data for the returns of the assets and the exchange rates for a selected
period of time, we run the optimization problem on GAMS model, and we get the optimal weights for each
asset. By determining the efficient frontiers and conducting a backing test for domestic and international
portfolios, we reach to a conclusion about the profitability of the investments. In the end, we interpret the
outputs, by analyzing the historical returns of the assets in tandem with the effect of exchange rates.
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2 Kinds of risk The term “risk” describes the uncertainty that the realized return of an investment will not be equal to the
expected return. The financial instruments contain various risks, according to their specific characteristics
and the transactions among them comprise high risks of reducing or losing the initial (or multiple) investing
capital. Thereby, the transaction with financial instruments is appropriate to investors who understand and
interpret both the movements of these instruments and the undertaken risks. There are several forms of risk
that affect the performance of a portfolio and the investors are trying to mitigate them.
2.1 Market risk: is the risk which can cause fluctuations both to the fair value and to the future cash flows of the financial
instrument. It is also common known as “Systematic risk” and is the probability of a loss as a result of
negative fluctuations in the market. For instance, a decrease in the price of equities, interest rates,
currencies, commodities, valuable metals and generally values that are traded in the market, can reduce the
invested capital. Market risk can affect the whole market and the investors can mitigate it by investing in a
variety of asset classes to delegate the possibilities.
2.2 Exchange risk: is the risk of the impact on evaluating an investment in foreign currency, due to a fluctuation of exchange
rates. Furthermore, we can define the Exchange risk as the risk that can be caused from the changes in
exchange rates, affecting the fair value or the future cash flows of a financial instrument. Once an investor
has to deal with assets that are exposed to foreign currencies, he is prone to unpredictable fluctuations of
the currency’s price.
2.3 Interest rate risk: contains the possibility the value of an investment to be undermined due to a change in interest rates. As
interest rate rise equity’s and bond’s prices fall and vice versa. . The relationship between the interest rate
and bond prices can be interpreted by opportunity risk. When an investor buys bonds, assumes that if the
interest rate increases, he will miss out the opportunity of purchasing the bonds with better returns. In that
case, the demand for the bonds with lower returns declines because of the new opportunities that appear.
The choice of new bonds with higher return rates that are issued, are now more attractive for the investors,
than the previous bonds that they hold. The interest rate risk is called “the source” of market risk as the
fluctuations in interest rates affect the market in total. In conclusion, it is a risk through fair value and future
cash flows can either be affected negatively or have intense fluctuations.
2.4 Credit risk: is the risk of one of the two parts to abrogate its obligation, causing loss to the other part. It is known that
a bond is a kind of loan, issued by a government or an organization. Although the risk of issuer’s default is
the greatest factor of setting the appropriate borrowing rate, usually the market can predict it, especially in
periods of growth. When an investor buys a bond or a kind of loan has expectations for a return. The return
is a summary of interest rate risk and credit risk. The interest rate risk is reflected in reference rate and the
credit risk in credit spread. The credit risk wasn’t able to be isolated until the appearance of credit options.
Through these instruments, in a market, credit risk was attainable to be transferred to the investors who had
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the willing to take it. (Through the composition of a bond and an asset swap, the credit risk could be
isolated).
The euro system has adapted rules and institutions for the assessment of credit risk among the countries.
The rating agencies (Moody’s, Standard & Poor’s, Fitch e.t.c.) have the duty to assess credit risk of the
countries and big companies, providing useful information to the investors. If a bond has a low rating (B or
C), the issuer has a high risk of default. Conversely, if it has a high rating (AAA, AA, or A), it's considered
to be a safe investment. Also, Central Banks like Deutsche Bundesbank, Banque de France, Banco de
Espana, Oesterreichische National bank provide detailed reports for the credit assessment.
2.5 Liquidity risk: stems from the lack of marketability of an investment that happens when a financial product cannot be sold,
as a result of not being attractive to the investors. In that case, the investor might be unable to convert an
asset into cash without giving up capital and income due to a lack of buyers or an inefficient market.
Usually, it is reflected either in bid-ask-spreads (the amount by which the ask price exceeds the bid price
for an asset in the market) or in big price movements. These facts depict uncertainty and insecure based on
the fact that a borrower cannot meet its short-term debt obligations. The liquidity risk is used from the
investors as a measurement of the level of risk within a company. In the level of financial institutions, their
sustainability depends on the ability to meet their debt obligations without realizing great losses.
2.6 Inflation risk: is the risk that inflation (the increase in the prices of a basket of selected goods and services over a period
of time) will erode the value and expected profit of an investment. When we interpret the performance of
an investment without taking into consideration the inflation, the outcome that turns out is the nominal
return. However, an investor should take into account the real return of its investment which is linked to
the purchase power. Bonds are the financial instruments that are most vulnerable to inflationary risk. Most
of the times, bonds obtain a fixed coupon rate that does not increase. Thus, if an investor buys a bond that
pays a fixed interest rate, but inflation rises more than the interest rate, then the investor faces losses. In
case this situation will go on until the maturity of the bond, the holder will lose more purchasing power,
regardless of how safe investment it is.
2.7 Settlement risk: is the risk that one party will fail to deliver the terms of a contract with another party at the time of
settlement. It can happen when one side fails to fulfill its obligations either by not delivering the underlying
asset or by not paying the cash value of the contract. The organized markets try to confine this kind of risk
by adopting the appropriate procedures. In not structured markets (out of money market) the investor faces
the credit risk of the counterparty.
2.8 Sovereign risk: contains the possibility a central bank will apply exchange rules that will affect negatively the worth of the
FOREX (Foreign Exchange) contracts that have been done by foreign investors. A FOREX contract is a
kind of currency transaction, where the two parties make an agreement in order to exchange two designated
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currencies at a specific future time. It is a common way for the byer to protect from currency price’s
fluctuations. Foreign exchange traders have to deal with the danger that a central bank will alter its monetary
policy and as a result will affect currency trades. Another factor that causes sovereign risk is the political
instability that takes place when a government refuses to adhere to the payments agreement of its sovereign
debt.
3 Risk measures and mathematical formulations
Portfolio optimization problem is an important topic since the pioneering Markowitz work on optimal
portfolio selection. Specifically, an investor is interested in determining the optimal combination of n risky
assets in such a way that the obtained portfolio has minimal risk and maximal expected gain. Although the
idea of risk seems to be intuitively clear, it is difficult to formalize it. There is an efficient way to measure
and quantify risk in almost every single market. However, each method is deeply associated with its specific
market and cannot be used directly to other markets. For instance, the most useful way to measure risk on
an equity portfolio is the volatility of returns, whereas for a government bond is the interest rate risk.
Furthermore, for portfolios of options, the risk can be measured by the rate of the change in price when
only one of the parameters changes (like the underlying asset, volatility, time to maturity). These indices
are suited to measure the very specific types of risk related to a specific financial instrument.
Based on investors preferences and risk tolerance several alternative formulations are available that will be
analyzed in the current thesis. Prior to presenting the various risk measures and the mathematical
formulation of them, it is necessary to set the “scenario generation” which is the base of the analysis.
Scenario generation:
Scenarios are used in order to include and measure all the disparate sources of risk that affect the
performance of portfolio. Each scenario depicts a range of different parameters and values that react on the
portfolio. Thus, is a way to represent uncertainty that stems from the variations of the included parameters.
Probabilities Scenarios 𝑟1 𝑟2 ….. 𝑟𝑛
𝑃𝑟𝑜𝑏1 𝑆1 𝑟11 𝑟12 ….. 𝑟1𝑛
𝑃𝑟𝑜𝑏2 𝑆2 𝑟21 𝑟22 ….. 𝑟2𝑛
𝑃𝑟𝑜𝑏3 𝑆3 𝑟31 𝑟32 ….. 𝑟3𝑛
…... ….. ….. ….. ….. …..
𝑃𝑟𝑜𝑏𝑇 𝑆𝑇 𝑟𝑇1 𝑟𝑇2 ….. 𝑟𝑇𝑛
Let us assume that we have an universe of n instruments (i = 1,2,…, n)
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�̃� = (𝑟1̃, 𝑟2̃, 𝑟3̃, … , 𝑟�̃�)𝑇, where �̃� is the vector of the random instrument’s return, which are unknown
at the time of portfolio selection and are treated as random variables.
𝐸(�̃�) = �̅� = (𝑟1̅, 𝑟2̅, 𝑟3̅, … , 𝑟�̅�)𝑇 , where E(�̃�) is the Expected return or the mean of a random
instrument.
𝑥 = (𝑥1, 𝑥2, … , 𝑥𝑛)𝑇 , where x is the allocation of an investor’s budget in the assets.
∑ 𝑥𝑖 = 1𝑛𝑖=1 (budget constraint)
𝑋 = {𝑥 ∶ 𝑥𝑇 1 = 1 , 𝑥 ≥ 0}, which is the basic portfolio constraints in vector notation by using the
conformable vector 1 = (1,1, … ,1)𝑇of ones.
𝑅(𝑥, �̃�) = 𝑥𝑇 ∙ �̃� = ∑ 𝑥𝑖 ∙ 𝑟�̃�𝑛𝑖=1 , where 𝑅(𝑥, �̃�) is the portfolio’s uncertain performance at the end
of the holding period.
𝛺 = {𝑠|𝑠 = 1,2, … , 𝑇} is the finite set of discrete scenarios.
𝑃𝑟𝑜𝑏𝑠 > 0 is the probability under a particular scenario (s) take the returns �̃�
𝐸[𝑅(𝑥, �̃�)] = 𝑥𝑇 ∙ �̅� = ∑ 𝑥𝑖 ∙ 𝑟�̅�𝑛𝑖=1 , where 𝐸[𝑅(𝑥, �̃�)] is the Expected performance of portfolio.
𝐹(𝑥, 𝑛) = 𝑃𝑟𝑜𝑏(𝑅(𝑥, �̃�) ≤ 𝑛), where F(x,n) is the market portfolio.
According to the above table:
𝑅(𝑥, 𝑟1) = ∑ 𝑥𝑖 ∙ 𝑟𝑖1𝑛
𝑖=1
𝐸[𝑅(𝑥, 𝑟)] = ∑ 𝑃𝑟𝑜𝑏(𝑠) ∙ 𝑅(𝑥, 𝑟𝑠)𝑠𝑇𝑠=1
We use the function 𝜑(𝑥, 𝑟) as a risk measure. The choice of the risk measure is related to the investor’s
profile and his preferences. Then for a certain minimal expected portfolio return μ, the 𝜑-efficient portfolio is the result of the below problem solution:
min 𝜑( 𝑥, 𝑟)
s.t. 𝐸[𝑅(𝑥, 𝑟)] ≥ 𝜇
Axiomatic investigation of risk measures was suggested by Artzner et al. [1]. Rockafellar [2]defined a
function as a coherent risk measure in the extended sense if it meets the following axioms:
1. Sub-additivity: 𝜑(�̃� + �̃�) ≤ 𝜑(�̃�) + 𝜑(�̃�)
2. Homogeneity: 𝜑(𝜆 ∙ �̃�) = 𝜆𝜑(�̃�)
3. Monotonicity: 𝜑(�̃�) ≤ 𝜑(�̃�) ⇒ �̃� ≪ �̃� (we prefer �̃� than �̃�)
4. Risk-free condition:(�̃� − 𝑛 ∙ 𝑟𝑓) = 𝜑(�̃�) − 𝑛 , where 𝑟𝑓 is the riskless rate.
A function is called a coherent risk measure if it satisfies axioms 1,2,3,4.
It’s vital for the risk measure to satisfy the first axiom (sub-additivity) because it allows the
diversification. Diversification is a risk management strategy that mixes a wide variety of
investments within a portfolio. Thus, a diversified portfolio has a significant lower risk because of
the limited exposure to any single asset or risk. Last but not least, sub-additivity and homogeneity
ensure that the risk measure function is convex, which is consistent with the theory of risk aversion.
After setting the scenario generation, the next section defines the risk measures and their basic properties
through their mathematical formulation:
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3.1 Variance: The Mean-Variance model, which was introduced by Markowitz in 1952 and he published the article
Portfolio selection in Journal of Finance 1952 [3], laid the foundations of the contemporary risk
management. The model presents two vital parameters with regard to portfolio optimization problems:
Expect return and risk. Portfolios have to tackle the problem of how to allocate wealth among several assets.
However, an investor always prefers to maximize the returns of the portfolio and concurrently to make the
risk as small as possible. Nevertheless, a high return always accompanied with a higher risk. That problem
is one of the important research fields in modern risk management. Markowitz applied variance (or standard
deviation) as a measure of risk and the mathematical formulation of it is:
𝜎2 = ∑ ∑ 𝑤𝑖𝑤𝑗𝜎𝑖𝑗
𝑁
𝑗=1
𝛮
𝑖=1
In that case, in order to make the notation simpler we introduce weights, where 𝑤𝑖 is the proportion of the
portfolio held in assets 𝑖 (𝑖 = 1, … , 𝑁) and 𝑤𝑗 is the proportion of the portfolio held in assets 𝑗 (𝑗 = 1, … , 𝑁).
Also, 𝜎𝑖𝑗 is the covariance between assets between 𝑖 and 𝑗.
The demise of the risk in a portfolio implies a low variance(𝜎2). Hence, the fundamental problem of a portfolio requires to minimize the variance with respect to a fixed expected return �̅�. This is known as the Markowitz model [4]
min ∑ ∑ 𝑤𝑖𝑤𝑗𝜎𝑖𝑗
𝑁
𝑗=1
𝛮
𝑖=1
s.t. ∑ 𝑤𝑖�̅�𝑖 = �̅�𝛮𝑖=1
∑ 𝑤𝑖 = 1
𝑁
𝑖=1
Alternatively, we want to maximize the portfolio return given constraints and given the portfolio
volatility:
𝑚𝑎𝑥 ∑ 𝑤𝑖 �̅�𝑖
𝑁
𝑖,𝑗=1
s.t. ∑ ∑ 𝑤𝑖𝑤𝑗𝜎𝑖𝑗 = 𝜎2𝑁
𝑗=1𝑁𝑖=1
∑ 𝑤𝑖 = 1
𝑁
𝑖=1
The volatility of the portfolio is a highly non-linear function of the weights, whereas most of the other
constraints are linear. Also, the portfolio return is linear in the weights. The volatility of the portfolio is a
highly non-linear function of the weights, while the majority of the rest constraints are linear, included the
portfolio return in the weights. By considering the mathematics of optimization we are able to select the
formulation of the problem. Thus, we prefer to minimize risk (the standard deviation or the variance of the
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portfolio), instead of maximizing return given non-linear constraints (for the level of risk). We can explain
that choice since it is easier to minimize a non-linear function with linear constraints than the alternative.
Also, it is important to mention the one-to-one relation between the standard deviation and the variance of
the assets. Because of that property we can minimize the variance and get the same results for the portfolio
allocation as would get if we minimized the standard deviation. Furthermore, another reason we prefer to
minimize the risk of the portfolio is that variance is a quadratic function and several convenient
mathematical methods exist in order to optimize them. One of them is called the method of the Langrangean
multipliers.
After presenting the portfolio optimization problem, using the variance of a portfolio as a risk measure, it
is important to mention the theoretical problems of Markowitz’s approach. Many researchers do not blindly
depend on the Mean-Variance model of Markowitz as it is based on the assumptions that (for a risk averse
investor):
the distribution of the rate of return is multivariate normal.
the utility function of is a quadratic function of the rate of return.
However, none of these assumptions applies in practice, leading the researchers and investors to develop a
plethora of models in order to tackle the Mean-Variance model’s defects. Thus, a lot of improved models
from computational and theoretical view have been made. In the current thesis a plethora of these models
are presented below.
3.2 Value at Risk (VaR): Information about future return of a portfolio is unknown at the time of portfolio allocation, so every
decision brings some risk. The Value at Risk is an important measure of the risk exposure of a given
portfolio. It is now the main tool in financial industry and part of regulatory mechanism.
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Let X be a random variable with the cumulative distribution function: 𝐹𝑋(𝑧) = 𝑃{𝑋 ≤ 𝑧}.X may have meaning of loss or gain, but in the particular situation X has meaning of loss.
The VaR of X with confidence level α ∈[0, 1] is:
𝑉𝑎𝑅(𝑥, 𝑎) = min{𝑢|𝐹(𝑥, 𝑧) ≥ 1 − 𝑎} = min{𝑢|𝑃[𝑅(𝑥, �̃�) ≤ 𝑢] ≥ 1 − 𝑎}, where 𝑎=95%(or 99%)
The VaR of a portfolio is defined as the maximum loss that will occur over a given period at a given
probability level. Since we are interested in the most unfavorable outcomes, we would like to measure the
level of losses which will not be exceeded with some fixed probability. The critical level typically used for
this fixed probability is either 5% or 1%.
Historically, VaR was introduced by financial companies in the late 80’s. Since then this approach has
become popular among practitioners, the academic community and – most significantly – among central
banks.
The greatest advantage of VaR is the recognition and the acceptance that has as a widely accepted standard.
Also, it is a good basis for comparison and measurement of risk between portfolios and investments, since
the industry and the regulators agreed to use it as the primary risk management measure. The two valuable
factors of the success of the current model is its simplicity in use and the stability and the consistency of
the results.
However, VaR risk measure has a grave disadvantage since it does not control scenarios exceeding VaR.
More specifically, VaR does not take into account the extreme tails of the distribution that contain large
losses with very low probability (less than(1 − 𝑎) ∙ 100%. Thus, the indifference of VaR to extreme
negative returns may be an undesirable property, allowing the investor to take extraordinary and undesirable
risks with a very small chance. Although this indifference of VaR hides a risk for the investors, it may have
a positive result in statistics as its estimators are robust and are not affected by excessive losses. Particularly,
these huge losses may “confuse” the statistical estimation procedure and produce misleading results.
Furthermore, VaR model has another important drawback that affects seriously the estimated risk.
Specifically, the model does not consider the property of sub-additivity which is a vital condition for a risk
measure to be coherent. The lack of sub-additivity depicts that VaR of two different investment portfolios
may be greater than the sum of the individual VaRs. As a result, the property of “diversification” is not
satisfied, and that fact limits the use of VaR.
Last, VaR is relatively difficult to optimize due to be a non-convex discontinuous function and exhibits
multiple local extrema. Although significant progress has been made in this direction, efficient algorithms
for solving such functions are lacking.
In conclusion, VaR is not a coherent measure of risk due to the facts below:
It does not take into consideration the extreme tails of the distribution that entail big losses.
It does not urge and, indeed, sometimes prohibit diversification.
It is difficult to optimize it.
In order to deal with these disadvantages and to avoid large errors in the final decision, an alternative risk
measure was introduced.
3.3 Conditional Value at Risk (CVaR): CVaR is a coherent risk measure that estimates the mean of the beyond-VaR tail region, which may be
widely used in the optimization of the portfolio. VaR answers the question: what is the maximum loss with
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the confidence level 𝛼 ∙ 100% over a given time horizon? Thus, its calculation reveals that the loss will exceed VaR with likelihood(1 − 𝛼) ∙ 100%, but no information is provided on the amount of the excess loss, which may be significantly greater. By contrast, CVaR is considered a more consistent measure of
risk than VaR. CVaR supplements the information provided by VaR and calculates the quantity of the
excess loss.
Below, CVaR is defined as the conditional expectation of portfolio returns below VaR for continuous
distributions, which measures the expected value of the (1 − 𝑎) ∙ 100% lowest portfolio’s returns:
𝐶𝑉𝑎𝑅 = 𝐸[𝑅(𝑥, �̃�)|𝑅(𝑥, �̃�) ≤ 𝑉𝑎𝑅]
However, the formula does not provide a coherent risk measure for discrete distributions as it gives a non-
convex function in portfolio positions. Furthermore, another drawback in case there is a discrete distribution
is the lack of a vital property that must be applied, sub-additive. In order to face these problems, we define
CVaR model for general distributions with the formula:
𝐶𝑉𝑎𝑅 = (1 −∑ 𝑃𝑟𝑜𝑏𝑠
{𝑠 ∈ 𝛺|𝑅(𝑥, 𝑟𝑠) ≤ 𝑧}1−𝑎
) ∙ 𝑧 +1
1−𝑎∙ ∑ 𝑃𝑟𝑜𝑏𝑠𝑅(𝑥, 𝑟𝑠){𝑠 ∈ 𝛺|𝑅(𝑥, 𝑟𝑠) ≤ 𝑧}
, where 𝑧 =
𝑉𝑎𝑅
The priority of each investor is to minimize the losses of his investments. In order to achieve that, as VaR
and CVaR tends more to the right part of the distribution than to the other direction where the bigger losses
exist, the investment is more profitable and appealing to the investors. To solve this risk management
problem:
max 𝑧 −1
1 − 𝑎∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑦𝑠
+
𝑠
𝑖=1
𝑠. 𝑡. 𝑅(𝑥, 𝑟𝑠) = ∑ 𝑥𝑖 ∙ 𝑟𝑖𝑠
𝑛
𝑖=1
𝐸[𝑅(𝑥, 𝑟𝑠)] = ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑅(𝑥, 𝑟𝑠)
𝑆
𝑠=1
≥ 𝜇
𝑦𝑠+ ≥ 𝑧 − 𝑅(𝑥, 𝑟𝑠)
𝑦𝑠+ ≥ 0
∑ 𝑥𝑖 = 1𝑛𝑖=1 , 𝑥𝑖 ≥ 0 ∀ 𝑖=(1, 2,…n)
Where 𝑧 = 𝑉𝑎𝑅 and 𝑦𝑠+is an instrumental variable. Also, we define the expected portfolio return with 𝜇.
At this point, it is crucial to define the instrumental variable which is a tool for the above optimization
problem. Hence, for every scenario 𝑠 ∈ 𝛺:
𝑦𝑠+ = max[0, 𝑧 − 𝑅(𝑥, 𝑟𝑠)]
This variable turns 0 when the return of the portfolio exceeds VaR for a particular scenario. In contrast,
when the portfolio return is below VaR the variable equals to the difference between VaR and the realized
performance.
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Through the above maximization problem, the “CVaR-efficient frontier” is generated, which is the result
of the CVaR’s optimization. It applies on each expected return 𝜇, related to each discrete scenario. Concurrently, apart from the “CVaR-efficient frontier” the VaR is calculated, providing a completed view
from the whole tail of the returns distribution which depicts all the chances, even the lowest ones, to realize
losses. At the end, the investor is informed about the allocation of their budget and the structure of their
portfolio since the optimized portfolio is produced.
The use of CVaR, as risk measure, is common in a lot of industries due to its properties that turn it into a
consistent and a coherent measure of risk. CVaR is a convex and continuous function, allowing the
optimization to be done quite efficiently. Also, its popularity is based on its property to take into account
and control scenarios exceeding VaR, on the extreme tails of the distribution. It allows the investors to
protect from the undesired risk which can hide substantial losses.
The problem of choice between the two risk measures (VaR and CVaR), especially in financial risk
management, has been quite popular in academic literature. The differences in mathematical properties,
simplicity of optimization procedures and the accuracy in calculation and control the risk are the main
factors that affect an investor’s decision. VaR has achieved popularity as an appropriate risk measure in a
plethora of engineering applications, including the financial sector. On the other hand, CVaR is heavily
used in the insurance industry since the achievement of profits is related to the effective minimization of
risk. However, there is no doubt that both are widely popular for risk measure so in the academic research
as in the economy.
3.4 Mean Absolute Deviation (MAD) The MAD model is a risk measure which is widely used to solve large-scale portfolio optimization
problems. It was introduced by Konno and Yamazaki (1991) due to the standard Mean-Variance model
(MV) of Markowitz couldn’t deal with huge models. The innovative model uses the absolute-deviation of
the rate of return of a portfolio instead of the variance to measure the risk. Although these two measures
are almost the same from mathematical view, they have significant differences from computational view.
Specifically, MAD model can be optimized by a linear program, whereas MV leads to a convex quadratic
programming problem. The former can be solved distinctively faster than the latter and this is one of the
main reasons we prefer MAD model.
The definition of MAD model describes risk as the mean absolute deviation of portfolio return from its
expected value. As it is conspicuous from the mathematical formulation:
𝑀𝐴𝐷(𝑥) = 𝐸{ |𝐸[𝑅(𝑥, �̃�)] − 𝑅(𝑥, �̅�)|}
When we refer to a discrete scenario set instead of a random asset’s return:
𝑀𝐴𝐷(𝑥) = ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ |𝐸(𝑅) − 𝑅𝑠|
𝑆
𝑠=1
We could define the MAD model as the average of the positive distances of each point from the mean. The
larger the MAD, the greater variability there is in the data. In order to optimize MAD, we need to solve the
following linear problem:
min ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑦𝑠
𝑆
𝑠=1
s.t. 𝑅(𝑥, 𝑟𝑠) = ∑ 𝑥𝑠 ∙ 𝑟𝑠𝑁𝑠=1
13
𝐸[𝑅(𝑥, 𝑟𝑠)] = ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑅(𝑥, 𝑟𝑠)
𝑁
𝑠=1
≥ 𝜇
∑ 𝑥𝑖 = 1𝑛𝑖=1 , 𝑥𝑖 ≥ 0∀ i=(1, 2,…,n)
𝑦𝑠 ≥ 𝐸[𝑅(𝑥, 𝑟)] − 𝑅(𝑥, 𝑟𝑠) ∀ 𝑠 = 1,2, … , 𝑆
𝑦𝑠 ≥ 𝑅(𝑥, 𝑟𝑠) − 𝐸[𝑅(𝑥, 𝑟)] ∀ 𝑠 = 1,2, … , 𝑆
𝑦𝑠 ≥ 0
Comparable to CVaR model, we introduce an instrumental variable to linearize the absolute value
expression. By setting the above conditions, we ensure that 𝑦𝑠 gathers the positive difference of the mean,
either a bit is higher or lower than this. In case 𝑦𝑠 < 0 then it equals 0. The notion we follow is the same as it was in CVaR model. By solving the linear problem, the MAD- efficient frontier is generated according
to the different values of the expected return 𝜇. For each expected return 𝜇 the investor manages the risk, since the output of MAD model is the allocation of an investor’s budget. Thereby, we have achieved to
solve large- scale portfolio optimization problems with a wider computational simplicity than using the MV
model and to build the efficient portfolio for each 𝜇, based on the MAD-efficient frontier. These models have been used in a plethora of portfolio optimization problems because of the groundbreaking method that
was introduced in the 90’s.
3.5 Tracking error models In the tracking error models, we introduce a different concept for allocation of the capital in comparison to
the previous ones. In this case, the professional money manager (institutional investor) has to structure the
portfolio according to a pre-specified benchmark. In other terms, the goal of a tracking error portfolio model
is to replicate, as close as possible, a given benchmark. Therefore, the portfolio returns must relate to the
performance of the imitated target. The allocation decision problem is based on the difference between the
manager's return and the benchmark return, the so-called tracking error. The tracking-error optimization
problems could be solved by finding a portfolio with maximum expected performance relative to the
benchmark for a given tracking error volatility. For the implementation, it is necessary to control the
deviations between the portfolio returns and the target. To achieve it, we impose an infinite penalty for any
deviations that are more than a user specified parameter 𝜖𝑅(𝑥, �̅�) below the target. The condition contains
both the cases where deviations being within 𝜖𝑅(𝑥, �̅�) below or above the benchmark and greater than it. Thus, we do not take into consideration returns that do not comply with that condition. The next step is to
set the mathematical formulation:
max ∑ 𝑝𝑠𝑅(𝑥, 𝑟𝑠)
𝑆
𝑠=1
s.t. 𝑥 ∈ 𝑋
𝑅(𝑥, 𝑟𝑠) ≥ 𝑔𝑠 − 𝜖𝑅(𝑥, �̅�), 𝑠 = 1,2, … , 𝑆
where 𝑔𝑠is the return of the benchmark for each scenario 𝑠 ∈ 𝛺.
It is conspicuous that the condition affects the downside risk which is defined with respect to the target.
The target may be set equal to the return of an alternative investment opportunity or to a fixed value and
thereby the target is the desired return we want the portfolio to achieve. Parameter 𝜖 is determined by the user and it is utilized to estimate the weights of each asset. In other words, the parameter is used to produce
14
the optimal portfolio. For that reason, it is essential to set an appropriate 𝜖 that is suitable with the data of
the problem. The goal is to find the parameter with which we will produce the optimal portfolio for each
scenario. In order to structure a portfolio close to the target, it is conducive to set the parameter equal to a
small number because as we can observe from the above condition a low 𝜖leads to a portfolio return closer
to the selected return we want to achieve. In conclusion, the parameter𝜖 depicts that the smaller it gets, the closer our portfolio approaches the benchmark.
3.6 Put/Call Efficient Frontiers Model The main notion of the model is to build up a portfolio that trades off the risk (downside) against the reward
(upside) given a benchmark (target). It is known that the portfolio’s reward is calculated through the
expected return of the portfolio. The main goal of that model is to create a portfolio that performs better
than the target for each scenario. In order to achieve that, it is crucial to measure both the portfolio’s
outperformance from the target and the underperformance from the target. The tools that will provide all
the needed information are two auxiliary variables counting the positive and the negative deviations of the
portfolio return from the benchmark. Thus, we introduce 𝑦+ in order to measure the portfolio returns and
𝑦− in order to captivate the risk:
𝑦+ = max[ 𝑅(𝑥, �̃�) − �̃�, 0]
𝑦− = max[ �̃� − 𝑅(𝑥, �̃�), 0]
These auxiliary variables have a crucial role as they distinguish the model from the other risk measures. As
it is conspicuous from the name of the model, the payoffs of the upside potential on the future portfolio
return are similar to a call option, having as a strike price the price of the target. In contrast, the payoffs of
the downside payoffs on the future portfolio return are similar to those of a short position in a put option.
In other terms, when the return of the portfolio is above the target, the gains are identical to the payoffs of
an exercised call option. The investor gains the difference between the performance of the portfolio and the
price of the benchmark. Obviously in that case, there is a zero- downside potential and the put option is not
exercised. On the contrary, when the portfolio return is below the target, the downside payoffs are identical
to those of a short position in a put option. Then, the investor gains the difference between the price of the
benchmark and the low portfolio performance. Again, in that case, the call option cannot be exercised as
the upside potential is zero. The following figure depicts the Put/Call Efficient Frontier Model:
𝑅(𝑥, 𝑟𝑠)𝑅(𝑥, 𝑟𝑠)
𝑔𝑠
Also, 𝑦+ is used to measure the upside potential of the portfolio to outperform the benchmark, while 𝑦−is used as a measurement of the downside risk of the portfolio. In discrete scenario setting we can express the
definitions of the auxiliary variables as systems of inequalities:
𝑦+𝑠 ≥ 𝑅(𝑥, 𝑟𝑠) − 𝑔𝑠, 𝑦+
𝑠 ≥ 0, ∀𝑠 ∈ 𝛺
𝑦+ 𝑦−
15
𝑦−𝑠 ≥ 𝑔𝑠 − 𝑅(𝑥, 𝑟𝑠), 𝑦−
𝑠 ≥ 0, ∀𝑠 ∈ 𝛺
The goal of the model is to produce portfolios that achieve the higher call for a given put. Under this
occasion, the investment is considered to have the greatest performance and the portfolio is called put/call
efficient. By definition, a portfolio is efficient when it offers the highest expected return for a defined level
of risk or the lowest risk for a given level of expected return. The mathematical formulation of the Put/Call
Efficient Frontier Model in discrete scenario setting is represented with a linear program for tracing the
efficient frontier.
max ∑ 𝑝𝑠𝑦+𝑠
𝑆
𝑠=1
s.t ∑ 𝑃𝑟𝑜𝑏𝑠𝑦−𝑠 ≥ 𝑤𝑆𝑠=1 , w: risk target
𝑦+𝑠 ≥ 0 ∀𝑠
𝑦−𝑠 ≥ 0 ∀𝑠
𝑦+𝑠 − 𝑦−
𝑠 − [𝑅(𝑥, 𝑟𝑠)] − 𝑔𝑠] = 0, ∀𝑠 ∈ 𝛺
The output of the maximization of the model is the portfolio of which the returns are above the target,
limiting at the same time the downside losses.
3.7 Expected Utility Maximization Model
The previous models optimized a pre-specified measure of risk against a pre-specified measure of reward,
such as the expected return of the portfolio. However, the investor’s choices and behavior have a vital role
in portfolio selection and in setting a target return. In order to take into account such a need in portfolio
optimization, we use the expected utility maximization model which allows the users to optimize according
to their own criteria when trading of risk and rewards. The current model bases on the Expected Utility
criterion which takes into consideration the risk preferences of an investor. It derives from the ability of a
concave curve to depict the preferences of a risk-averse investor. According to that criterion, we can classify
two mutually excluded investment plans because of their expected utility, since the criterion choose the one
with the highest expected utility.
Furthermore, the expected utility criterion can classify investment plans that have different risk, in spite of
having the same expected return. Although the same level of expected returns implies same utility for the
risk-averse investor, it would be wrong if we were indifferent between these two choices. Expected utility
gives a solution on that selection problem, as it considers the risk preferences of the investor.
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The above figure illustrates two investment plans having the same expected returns:
𝐸(𝑌 + 𝑍�̃�) =1
2(𝑌 − 𝑍𝐴) +
1
2(𝑌 + 𝑍𝐴) = 𝐸(𝑌 + 𝑍�̃�) =
1
2(𝑌 − 𝑍𝐵) +
1
2(𝑌 + 𝑍𝐵) = 𝑌
but also, different variance as plan B shows higher deviations than A does. However, both of them depict
the same utility for the expected returns:
𝑈[𝐸(𝑌 + 𝑍�̃�)] = 𝑈[𝐸(𝑌 + 𝑍�̃�)] = 𝑈(𝑌)
The conclusion is that the investor can distinguish which of the two investment plans appeals to his profile
only by using the expected utility criterion.
From a mathematical view, the ability of the criterion to classify the investment plans with different risks
and the same returns is attributed to the concavity of the utility function. Consequently, the expected utility
has different prices for two plans that have the same expected performance. For a concave utility function,
the higher expected utility of the plan with the lower deviation is consistent to the investing behavior of a
risk-averse investor. Such a behavior is caused because of that plan includes a risk of lower loss in
comparison to the alternative plan, since the returns of the latter have higher variance.
From the view of a portfolio, the expected utility criterion can be generalized for each scenario s, where
𝑠 ∈ 𝛺. Thus, the expected utility is defined as the expected average price of the utility levels for each
scenario:
𝑉[𝑆1, 𝑆2, … , 𝑆𝑁; 𝑃𝑟𝑜𝑏1, 𝑃𝑟𝑜𝑏2, … , 𝑃𝑟𝑜𝑏𝑁] = ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑈[𝑅(𝑥, 𝑟𝑠)]
𝑁
𝑠=1
While the models of the previous sections are specialized to produce frontiers of efficient portfolios –from
which the user has to select one- the expected utility maximization model allows the user to choose a unique
portfolio that maximizes the user’s utility function. The maximization problem is presented as follows:
𝑈(𝑌 − 𝑍𝐵)
𝑌 − 𝑍𝐵
𝑈(𝑌 + 𝑍𝐵)
𝑌 + 𝑍𝐵
𝑈(𝑌 + 𝑍𝐴)
𝑌 + 𝑍𝐴
𝑈(𝑌 − 𝑍𝐴)
𝑌 − 𝑍𝐴
𝑈(𝑌)
𝑌
𝐸[𝑈(𝑌 + 𝑍�̃�)] =1
2⁄ 𝑈(𝑌 − 𝑍𝐵) +1
2⁄ 𝑈(𝑌+𝑍𝐵)
𝐸[𝑈(𝑌 + 𝑍�̃�)] =1
2⁄ 𝑈(𝑌 − 𝑍𝐴) +1
2⁄ 𝑈(𝑌+𝑍𝐴)
𝑈(𝑌)
𝑌
17
max ∑ 𝑃𝑟𝑜𝑏𝑠 ∙ 𝑈[𝑅(𝑥, 𝑟𝑠)]
𝑆
𝑠=1
s.t 𝑥 ∈ 𝑋
𝑅(𝑥, 𝑟𝑠) = ∑ 𝑥𝑖𝑟𝑖𝑠
𝑛
𝑖=1
∑ 𝑥𝑖 = 1
𝑛
𝑖=1
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4 International Diversification – Does it pay to go internationally?
In this part, we utilize the available literature and models to determine the benefits from investing overseas.
Specifically, we investigate how the benefits of international portfolio diversification differ across the
countries, taking into consideration a plethora of factors that could affect the result. Also, the analysis
includes the short selling constraints in investments that are imposed in many developing countries to
control capital flows. The outputs appear to be statistically significant and provide strong evidence about
the advantages of international diversification.
The first research we are going to analyze comes from Joost Driessen and Luc Laeven [5] who carried out
an investigation about how the benefits of international portfolio diversification differ across countries from
the perspective of a local investor. Particularly, they investigate for a large cross-section of countries with
both large and small stock markets, what benefits occur by adding international stock investment
opportunities for a domestic investor that invests in local stocks only. They also measure the economic size
of these benefits. Although the literature on international portfolio diversification takes a U.S. perspective,
this paper takes the viewpoint of domestic investors in many different countries, since the benefits of
investing abroad for investors in countries with less well-diversified and developed stock markets as those
in the United States are likely to benefit more from investing abroad than U.S investors do. Also, the writers
consider both the case of frictionless markets, where the investors don’t face any costs to short assets, and
the case where short selling constraints prevail especially in developing countries. Furthermore, the results
are calculated both in local currencies since we consider a local investor that uses it, and in U.S. dollar as
in practice, investors in many counties use it as their base currency.
The analysis bases on two types of diversification opportunities appeared for an investor that currently
invests in the local equity index:
A. Regional diversification: A regional equity index for the country’s region
B. Global diversification: MSCI indices for the US, Europe and Far East.
The reason for distinguishing between these two scenarios is that it is interesting to see how much of the
international diversification benefits can be achieved by investing regionally and globally.
To measure the benefits of diversification for domestic investors, we use Markowitz’s notion about the
standard mean-variance, assuming either a mean-variance utility function for the investor, or normally
distributed asset returns. The concept is to examine if an addition in portfolio’s assets will shift the mean-
variance frontier.
In order to measure the economic size of diversification benefits, we use the increase in Sharpe ratios and
in expected return that is obtained when adding assets to the investment set. An alternative way to measure
it is through the expected return. If the expected return is increased when adding to the portfolio the new
𝑁 assets given the same (or lower) variance as for the optimal portfolio of 𝐾 benchmark assets, then the
diversification benefits are economic significant.
To calculate the statistical significance of the diversification possibilities, they use the regression tests for
mean-variance spanning developed by Huberman, Gur and Kandel (1987) [6] , DeRoon, Nijman, and
Werker (2001) [7], and DeRoon, Frans and Nijman (2001) [8]. In that test, it is examined whether adding
𝑁 new assets to a given set of K benchmark assets leads to a significant shift in the mean-variance frontier.
In the research, the domestic index is the K benchmark assets, so that K equals to 1, and the 𝑁 additional
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assets are given by asset sets A or B (either regional or global diversification is implemented). In case of
frictionless markets, the test can be performed using the regression:
𝑟𝑡+1 = 𝑎 + 𝛽𝑅𝑡+1 + 𝜀𝑡+1
Where 𝑟𝑡+1 is a 𝑁-dimensional column vector with the 𝑁 returns on the additional assets, 𝑅𝑡+1 is a 𝐾-
dimensional return vector for the 𝐾 benchmark assets, 𝑎 is a 𝑁-dimensional constant term, 𝛽is a 𝑁 by 𝐾
matrix with coefficients and 𝜀𝑡+1is a 𝑁-dimensional vector with zero-expectation error terms.
The null hypothesis that the 𝐾 benchmark assets span the entire market of all 𝐾 + 𝑁 assets is equivalent to the restrictions:
𝑎 = 0 𝑎𝑛𝑑 𝛽𝜄𝛫 = 𝜄𝛮
where 𝜄𝛮 is a 𝑁-dimensional vector with ones.
If the equations hold, the return on each asset that we add to the portfolio can be decomposed into the return
on a portfolio of benchmark assets and a zero-expectation error term that is uncorrelated with the benchmark
portfolio return. Hereby, from the view of mean-variance framework, such an additional asset can affect
the variance of the portfolio return and not the expected return and the investor will not include the
additional asset in his strategy. In other terms, if the spanning hypothesis holds, the optimal mean-variance
portfolio consists of the 𝐾 benchmark assets.
Also, it is vital to analyze how global diversification benefits change over the time. They use the country
risk measure to interpret the correlations of local returns with global indices and the variances of local
indices.
Finally, as far as the data concerns, for developed countries the writers use the stock market index developed
by Morgan Stanley Capital International (MSCI) and for developing countries they use the S&P/IFC Global
Index from Standard & Poor’, on a monthly base for a period 1985 to 2002. The research includes 23
developed markets and 29 developing markets. Also, they download country risk ratings for each country
s reported by the International Country Risk Guide maintained by the Political Risk Services group and
they use the ICRG composite index as a general measure of country risk. The rating ranges from 0 to 100,
with a higher rating implying less country risk. Furthermore, they download monthly currency data from
Datastream in order to returns in U.S.D. terms and in terms of the local currency.
4.1 Empirical Results First of all, they estimate both increases in Sharpe ratios and in expected returns given variance, when a
regional index or three global MSCI indices are added to a local stock index portfolio, in order to measure
the diversification benefits for the local investors of the 7 regions (Latin America, Asia, Eastern Europe,
Middle East & Africa, Europe 16, Pacific, North America. The results for each region are divided into two
types of diversification opportunities: Regional Diversification and Global Diversification. Also, the Sharpe
ratios are calculated for each country both with and without short-selling constraints and in terms of local
currency and U.S.D. returns. Short-selling constraints are either applied to all countries or only to
developing countries. The following table reports the results summarized by averaging country-by-country
results within each region.
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Table 1: Increase in Sharpe ratio and Expected return by region
It is conspicuous from the Table 1 that the regional diversification benefits appear largest on average for
countries in Eastern Europe, even when allowing for short selling constraints. For instance, the expected
return for Eastern European countries of investing in the region is 0.291% per month (3.492% per year)
higher than the return of the investment in the local market only, in U.S.D. terms, even when short selling
constraints prevail.
For the majority of the investors of the seven regions, the global diversification benefits are significantly
higher than the regional diversification benefits, according to both of Sharpe ratio and Expected returns,
with the largest benefits appear for the countries in Eastern Europe. These figures suggest that the economic
size of global diversification benefits is large for most countries, since the increases in Sharpe ratio and in
expected returns implies economic significance of the diversification benefits.
In a plethora of developing countries, short-selling constraints are imposed due to the government restricts
the investors to invest in their home country, which is potentially costly for these investors, since these
restrictions reduce the diversification possibilities. Therefore, the writers use short-selling constraints in
order to have more realistic results. However, it is obvious from the table that the results of global
diversification benefits are not affected significantly when assuming short sales constraints in developing
countries, due to for almost all countries it is optimal to invest a part of the total asset portfolio to the local
stock market index. But when we assume short sails constraints for all countries, which is a less realistic
assumption than the imposed constraints in developing countries, there is a substantial decrease in the
benefits of global diversification for the majority of the countries, especially when diversification benefits
are measured in terms of increasing in expected returns. Nevertheless, the global diversification benefits
for most of the regions are substantial and statistically significant as they were without short sales
constraints.
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Finally, as far as the currency concerns, there isn’t a significant difference in the results of diversification
benefits when we interpret them either in U.S.D. terms or in the local currency terms.
The next step is to calculate the statistical significance of the results by using the regression tests for mean-
variance spanning. Table 2 depicts the results of the spanning tests on a regional basis. Specifically, it
includes the percentages of cases in which the spanning hypothesis is rejected.
Table 2: Rejection of Spanning Tests by Region
When the global diversification takes place, the spanning hypothesis is rejected for all countries at 1%
confidence level, both when the short sales constraints are imposed to the investors in developing countries
and when the investors do not face any constraints at all. That means the global diversification benefits are
statistically significant for domestic investors around the world, due to tiny p-values that force us to reject
the spanning hypothesis for all confidence levels.
However, when we introduce short-selling constraints for all countries, which is not a realistic scenario, the
spanning hypothesis is not rejected for several countries in the regions, proving that the constraints
disappear both the regional and global diversification benefits for a plethora of countries. Despite that fact,
there are still many countries that gain benefits from diversification.
In the next section, the writers aim to interpret which factors affect the benefits of international
diversification in the various countries. Particularly, they explore whether the country risk and other country
characteristics have an effect on the gains from international diversification. In order to test it, they use a
cross-country regression as a mean to recognize the factor that affects the benefits. They use as dependent
variable the difference between the Sharpe ratio of the global portfolio (including the local index) and the
Sharpe ratio of the local index so as to measure the economic benefits for a mean-variance investor who
invests not only in the local but also in the global market. The higher figure this variable indicates the larger
benefits the investor gains. The regression is the following:
𝐼𝑆𝑅𝑖,𝑡 = 𝑎 + 𝑏𝐼𝐶𝑅𝐺𝑖,𝑡 + 𝑐𝑆𝑀𝐶𝑖,𝑡 + 𝑑𝑇𝑟𝐺𝐷𝑃𝑖,𝑦 + 𝑒𝑃𝐶𝐺𝐷𝑃𝑖,𝑡 + 𝜀𝑖,𝑡
Where 𝐼𝑆𝑅 is the Increase in Sharpe Ratio, 𝑎 is a constant, 𝐼𝐶𝑅𝐺 is the country risk rating, 𝑆𝑀𝐶 is the
Stock Market Capitalization in U.S.D, 𝑇𝑟𝐺𝐷𝑃 is the trade to GDP, 𝑃𝐶𝐺𝐷𝑃 is the private credit to GDP
22
and 𝜀 is the error. The dependent variables are calculated using stock market return and interest rate data
for the period 1985-2002.Generally, these variables proxy for the level of institutional development in the
country. It is expected that more development countries benefit less from international portfolio
diversification. The results of the regression are presented in Table 3:
Table 3: Cross-country differences in the benefits of foreign diversification
In comparison to all dependent variables used in the estimation, they found only the Country risk rating to
be statistically significant for the increase in the Sharpe ratio variable. Also, a vital output is that the
international diversification benefits are larger for countries with higher country risk. Nothing special
changes when we interpret the results in local currency terms than in U.S.D. terms and the short selling
constraints do not affect the results. To control robustness, they excluded the region of East Asia and
Russian due to the sample period covers both the Asian and Russian debt crisis of 1997 and the results did
not change significantly.
Except for the paper of Driessen and Laeven, another research carried out by Wan-Jiun Paul Chiou [10]
analyzes the benefits of international diversification by using two different measures in order to estimate
them. Specifically, the writer utilizes two straightforward measures: the increase in risk-adjusted premium
by investing in the maximum risk-adjusted return (MRR) portfolio and the reduction in the volatility by
investing the minimum variance portfolio (MVP) on the international efficient frontiers. The mathematical
formulation of the increment in risk-adjusted premium requires setting the maximum risk-adjusted return
(MRR). Thus, if there is no investment restriction neither for the developing nor the developed markets:
𝑀𝑅𝑅 = max {(𝑤𝑝
𝑇𝜇
𝑤𝑝𝑇𝑉𝑤𝑝
)
12⁄
|𝑤𝑝𝑇 ∈ 𝑆}
Where 𝑆 is the set of all real vectors 𝑤 that define the weights such that 𝑤𝑇1 = 1. The mean and variance-covariance of asset returns can be expressed as a vector 𝜇 and a positive definite matrix 𝑉, respectively.
From the above maximization problem, we expect to take the weights of the various assets in the portfolio.
However, the investor takes into consideration various constraints before investing in a foreign market and
the mathematical formulation can be obtained by the following:
23
𝑀𝑅𝑅 = max {(𝑤𝑝
𝑇𝜇
𝑤𝑝𝑇𝑉𝑤𝑝
)
12⁄
|𝑤𝑝𝑇 ∈ 𝑃𝐽}
Where 𝑃𝐽 = {𝑤𝑝 ∈ 𝑆 ∶ 0 ≤ 𝑤𝑖 ≤ 𝐽𝑤(𝐶𝑎𝑝)𝑖 ≤ 1, 𝑖 = 1,2, … , 𝑁}, 𝐽 is any number greater than one and
𝑤(𝐶𝑎𝑝)𝑖 is the weight of the market value of each country 𝑖. By setting the MRR, the greatest increment of unit-risk performance brought by international diversification, for the domestic investor, is:
𝛿𝑖 = 1 −𝑅𝑅𝑖
𝑀𝑅𝑅
Where 𝑅𝑅𝑖 is the risk-adjusted performance of domestic portfolio in country 𝑖. The second measure used for the estimation of the international diversification benefits is the reduction in volatility. Elton and Gruber
(1995) support that the estimation of the expected returns may hide a difficulty for the investors and for
that reason they seek to minimize the variance of a portfolio. A common way to minimize the variance of
a portfolio is by obtaining the vector of weights of the minimum-variance portfolio (MVP), 𝑤𝑀𝑉𝑃. According to the research of Li et al. (2003) [12], the maximum decline in volatility of domestic portfolio
in country 𝑖, by international diversification is:
𝜀𝑖 = 1 − [𝑤𝑀𝑉𝑃
𝛵 𝑉𝑤𝑀𝑉𝑃𝑉𝑖
]
2
Where 𝑉𝑖is the variance of return of the domestic portfolio. The data used in this study are the U.S. Dollar-denominated monthly returns of the MSCI indices for 21 developed countries and 13 developing countries
from January 1988 to December 2004. The differences of the benefits of global diversification for the
developed and developing countries with various investment constraints are reported below:
24
Table 4: Portfolio weights and benefit from global diversification
The significant cross-market differences in risk-adjusted premiums, implies that domestic investors in the
countries of low performance can improve the mean-variance efficiency of their portfolios by optimally
allocating funds to overseas assets. According to Table 4, the countries where the investors can gain the
highest mean-variance efficiency by investing internationally are Japan, Philippines, Thailand, New
Zealand, Portugal, Indonesia and Korea. Also, local investors can eliminate their portfolio’s volatility by
investing in strategies of MVP. The general conclusion is that investments from developing markets have
substantial margins of benefits by investing internationally, in contrast to investors that come from
developed countries.
According to the MRRP index, 6 countries only are proposed to invest in, whereas the MVP suggests 11
countries. Furthermore, the weights in small capital markets are excessive, while in the major financial
markets are significantly low. Based on these two indices, an investor should invest 80% of fund in
Argentina, Chile, Denmark, Mexico and Switzerland, which represent only 4% of global market value and
20% in the US markets. The results comply with the outcomes of Driessen and Laeven, since both of them
depict that there are substantial benefits for the investors of developing countries by investing globally.
Finally, the writer sites in Table 5 the unconstrained efficient frontiers composed by global portfolios,
countries of different developmental stages and various regions.
25
Table 5: Efficient Frontiers without constraints
The different positions of international portfolios composed by various groups of countries depict different
impacts on mean-variance efficiency. Specifically, the efficient frontiers that derive from developed
countries, Europe and North America are of low risk and low return, while the efficient frontiers of
investments in emerging markets and Latin America are of high risk with high return.
In conclusion, the two researches provide useful information about the differed benefits of international
portfolio diversification across countries from the perspective of a local investor. The first finding is that
there are substantial benefits from investing both regional and global for domestic investors in both
developed and developing countries, even under the realistic assumption that investors cannot short sell
stocks in developing markets. However, the benefits of international portfolio diversification are larger for
developing countries than for developed ones. Concerning the factors that play a crucial role in
determination of the benefits, country risk is vital, since countries with higher country risk have a greater
potential benefit of global diversification. The results are statistically significant. Another output of the
investigation is the decreasing trend of diversification benefits across the countries, due to the
improvements in country risk over time.
26
5 Empirical Application
In this section we implement a portfolio optimization problem of an investor who invests firstly in domestic
market and secondly in international capital markets. The problem is applied on an investor based in
Germany which means that the first part of the empirical application refers to the German market only,
while in the second one two foreign markets are added. We picked the United States of America (U.S.A)
and Japan as the two countries where the investor can invest in, except for his country, because of the
different level of development of the their financial markets. The assets which the investor can select for
his portfolio are 1 stock index and 2 bond indices from each country. Thus, in the case of the domestic
portfolio optimization problem, the portfolio can be consisted of up to 3 assets, whereas the international
portfolio up to 9 assets.
In order to calculate and evaluate the performance of both domestic and international portfolio during the
selected past period, we implement a method called back testing. Back testing is a methodology for
assessing the viability of a strategy by discovering how it would perform using historical data. Particularly,
it displays the level in which a portfolio would have performed ex-post and as a result it provides strong
evidence about the outcome of the investment when implemented in reality. The notion is that a portfolio
that would have performed poorly in the past will probably perform poorly in the future, and vice versa.
Thus, in our sample we implement back testing in order to assess the performance of the optimal portfolio
during a long period of time and to come to a conclusion about the future results of that specific investment.
However, before implementing the back testing, it is necessary to calculate the optimal weights of each
asset that an investor should invest in. In order to achieve it, we run a CVaR model on GAMS system which
solves the portfolio optimization problem.1 Hence, by holding the optimal weights of the assets we can
export the real returns of the portfolio that were performed in the past.
Apart from the back testing we construct the efficient frontier for the domestic and the international
portfolio since it is a way to understand how a portfolio’s expected returns vary considering the amount of
risk taken. Specifically, the efficient frontier is the locus between the returns and the CVaR (risk) of all the
combinations of the portfolios that offer the highest expected return for a defined level of risk. Those
portfolios that lie below the efficient frontier are sub-optimal because they do not provide enough return
for risk assumed. The efficient frontier is a commonly known path for the investors to select investments
since it provides information for all the kinds of investors. According to efficient frontier, a risk-seeing
investor would select portfolios that lie on the right end of the efficient frontier, since there are located
portfolios that are expected to have a high degree of risk coupled with high potential returns. Contrarily,
risk-averse investors will select portfolios that lie on the left end of the efficient frontier.
Last, it is vital to interpret the factors that affect the asset’s optimal weighs coming from the CVaR’s model
solution, so as to understand the results from a financial point of view. The two main reasons in international
portfolio which contribute to the optimal level of each asset are the descriptive statistics of the indices’
returns and the exchange risk since the investor based in Germany has to deal with assets that are exposed
to U.S. Dollar and Yen. Apparently, in the case of domestic portfolio the exchange risk does not exist, due
to the fact that the investments share joint currency.
1 The GAMS (General Algebraic Modeling System) is a high-level modeling system for mathematical programming. It can formulate complex and large-scale optimization models in a notation similar to their algebraic notation. Gams is designed to solve linear, non-linear and mixed-integer optimization problems and thus it can use many different solvers.
27
5.1 Data Description As referred, we are considering 3 assets from each country (1 stock index and 2 bond indices). Particularly,
as far as Germany concerns, we focus on the DAX 30 index, on FTSE German Government Bond Index 1-
3 year sector and on FTSE German Government Bond Index 7-10 year sector. Furthermore, regarding to
U.S.’s financial markets, we examine S&P 500 index, the US FTSE World Government Bond Index
(WGBI) US, 1-3Y and the US FTSE WGBI US, 7-10Y. Last, in Japan, we take into consideration the
NIKKEI 225 Stock index, the FTSE Japanese Government Bond Index 1-3 year sector and the FTSE
Japanese Government Bond Index 7-10 year sector. We have collected historical prices for all the indices
starting from 31/12/2004 until 30/4/2020, on a monthly basis. Thus, we were able to compute the returns
for each asset through all the months of our period. Also, due to the investments in foreign currencies, we
have collected data regarding to exchange rates, specifically the exchange rate for U.S. Dollar (U.S.D.) to
Euro and the rate for Yen to Euro. Again, the exchange rates are available on a monthly basis, the same
periods as the 9 assets were.
5.2 Methodology In this part is made an analytical description of the steps that were followed using the GAMS model. First
of all, we selected the CVaR mathematical model in order to solve the optimization problem. CVaR is a
very popular coherent risk measure for optimal portfolio selection, because it goes beyond the information
revealed by VaR and thus it takes into account the distribution of the tail, where the losses can be huge
there.
5.2.1 Domestic Portfolio Efficient Frontier: As referred previously, we need to find the locus between the returns and the
standard deviation (risk) of all the combinations of the portfolios that offer the highest expected return for
a defined level of risk. The data which we import in the GAMS model are returns of the 3 German assets
for the last 10 years (120 observations). In the first step, we find the bottom end of the efficient frontier’s
curve, where the risk is at its lowest level as it relates to return. In order to calculate that point in GAMS
model, we maximize CVaR model. Thus, we get the expected return of the portfolio which lies on the left
end of the efficient frontier. In the second step, we export the upper limit of the efficient frontier which lies
on the right tail, by solving the problem of maximization of total returns on GAMS model. After these steps,
we hold the two extreme points on the efficient frontier’s curve which display the minimum and the
maximum expected returns. In order to determine the rest points, we separate the gap between the extreme
points in 8 equal spaces, getting 8 different expected returns. Then, we run the CVaR model 8 times, by
assuming every time one of the 8 expected returns that we calculated previously, as the expected portfolio’s
target return. Therefore, we locate the middle points and the efficient frontier is completed.
Back testing: In order to implement this dynamic test, for each specific month we run the model,
we have to adjust the historic data to the optimization problem. The period of implementation starts from
April 2017 until April 2020, which means that we solve the optimization problem 37 times (37 months).
For every month we run the model, we insert a fixed number of historical data which is 147, since we got
data for the returns of the assets for 147 months before April 2017 (the beginning of the test). Thus, the
sample of historical data is determined as the last 147 observations prior to the month we examine. For
example, when solving the optimization problem for June 2018, the inserted returns in the model are from
March 2006 until May 2018. After solving the portfolio optimization problem through GAMS model, we
get the optimal weights for the 3 assets that an investor should invest in, monthly. In order to evaluate the
28
performance of the investment for all the months of the test, we multiply the optimal weights with the
realized returns in the specific month we examine. We repeat that procedure for all the tested months and
in the end, we have created the monthly gains or losses of the portfolio and by summarizing them we get
the final performance of the portfolio. Therefore, we are able to evaluate the return of a portfolio consisted
of 3 German assets the last 3 years, utilizing the CVaR model as a mean of selecting the optimal portfolio.
5.2.2 International Portfolio Efficient frontier: We implement the same methodology as we did in domestic portfolio, including
however the exchange rate of the foreign currencies. In order an investor from Germany to invest in
American and Japanese capital markets, it is necessary to convert euro in the currency that each asset can
be traded in. Thus, we need to find a mechanism for changing one currency into another, to be able to invest
in foreign assets. Our aim is to detect the return of each asset for all the months in the sample. The notion
is the following:
In order an investor from Germany to invest 1€ in a foreign country (USD or Yen), he has to convert it in
the specific currency. If the exchange rate is USD to Euro (or Yen), the type to convert 1€ in USD (or in
Yen) is:
1 ∗ 𝑒𝑐
𝑒𝑐: is the exchange rate in country c
Also, the performance of the asset 𝑖 in scenario 𝑛 in 1 month is:
1 ∗ 𝑒𝑐 ∗ (1 + 𝑅𝑖,𝑐𝑛 )
where 𝑅𝑖,𝑐𝑛 is the return of asset 𝑖 in scenario n.
In order to convert the performance in € (domestic currency):
1 ∗ 𝑒𝑐 ∗ (1 + 𝑅𝑖,𝑐𝑛 )
𝑒𝑐𝑛
where 𝑒𝑐𝑛 is the exchange rate in scenario 𝑛.
In conclusion, the performance of the asset 𝑖 is:
𝑒𝑐 ∗ (1 + 𝑅𝑖,𝑐𝑛 )
𝑒𝑐𝑛 − 1
However, this type is applied on our test since the exchange rates are USD to Euro and Yen to Euro. In case
the rates were expressed vice versa (Euro to Yen and Euro to USD), the performance of the asset 𝑖 is:
𝑒𝑐𝑛 ∗ (1 + 𝑅𝑖,𝑐
𝑛 )
𝑒𝑐− 1
By utilizing this type, we are able to create the returns for all the foreign assets on a monthly basis. Then,
we follow the same procedure as we did in domestic portfolio. The only different point is that the portfolio
is consisted of 9 assets, instead of 3 as it was in domestic portfolio. Hence, through the steps we
implemented in domestic portfolio related to maximization of CVaR model in GAMS, we draw the efficient
frontier of the international portfolio.
29
Back Testing: We repeat the methodology as we did in domestic portfolio by adding the 6 foreign
assets. In order to take into account the exchange rates that affect the returns of the investment in foreign
assets, we have to implement the steps that we did previously, so as to export the returns gained by an
investor from Germany. Hence, the notion of the back testing remains the same and we examine the
performance of the international portfolio, according to the returns performed in the past.
5.3 Empirical results We present below the results of the efficient frontier and back testing for the domestic and the international
portfolio.
5.3.1 Efficient Frontier
Table 6: Domestic Efficient Frontier
Figure 1: Efficient Frontiers
0
0,005
0,01
0,015
0,02
0,025
-0,14 -0,12 -0,1 -0,08 -0,06 -0,04 -0,02 0
International EfficientFrontier
DomesticEfficientFrontier
Efficient Frontiers
CVa
Expected Returns
Domestic Efficient Frontier
CVaR Expected Returns
Bottom end -0,00814 0,01897
m=0,001105 -0,00815 0,01899
m=0,00214 -0,00816 0,01901
m=0,003175 -0,00819 0,01903
m=0,00421 -0,00822 0,01905
m=0,005245 -0,00824 0,01907
m=0,00628 -0,00827 0,01909
m=0,007315 -0,0083 0,01911
Top end -0,03 0,01913
International Efficient Frontier
CVaR Expected Returns
Bottom end -0,00425 0,00007
m=0,001105 -0,00599 0,00111
m=0,00214 -0,00922 0,00214
m=0,003175 -0,01317 0,00318
m=0,00421 -0,01725 0,00421
m=0,005245 -0,02532 0,00525
m=0,00628 -0,04367 0,00628
m=0,007315 -0,0648 0,00732
Top end -0,125 0,00835
Table 7: International Efficient Frontier
30
In Table 6 and Table 7 are the results that we received from the GAMS model, after solving the optimization
problem using CVaR as risk measure. These results are represented in Figure 1 and by connecting all the
points we get the efficient frontier for each portfolio.
It is obvious from the graph that Domestic portfolio does not have significant volatility in expected returns,
since in the bottom end of the efficient frontier it is 0,01897 while in the top end it is 0,01913. The risk is
computed -0,00814 and -0,03 respectively, which shows that the German investor does not face a
substantial risk by investing in domestic portfolio. Furthermore, the efficient frontier of the international
portfolio displays that the maximum return an investor from Germany can gain is 0,00835 whereas the
minimum return is 0,00007. Also, the minimum price of CVaR (risk) is -0.125 and the maximum is -
0.00425.
Comparing the expected returns and the CVaR of the portfolios, we can observe that there is a positive
correlation between expected returns and CVaR. Specifically, in domestic portfolio there is higher CVaR
than in international one, due to the higher expected returns. The interpretation is attributed to the definition
of CVaR. Since CVaR quantifies the expected losses that occur beyond the VaR breakpoint, it is more
likely to be higher in a portfolio with larger expected returns. On the contrary, in international portfolio the
potential returns are low for a German investor and therefore they are associated with low levels of extreme
losses.
31
5.3.2 Back Testing
Domestic Portfolio
Portfolio’s
returns
Cumulative
portfolio’s returns
Optimal
Weight S1
Optimal
Weight S2
Optimal
Weight S3
Months -0,000293672 1 0,01342 0,98658 0
April ‘17 -0,000381804 0,999706328 0,01342 0,98658 0
May ‘17 -0,003693683 0,999324635 0,01342 0,98658 0
June ‘17 0,00096082 0,995633447 0,01342 0,98658 0
July ‘17 0,000610809 0,996590071 0,01342 0,98658 0
Aug. ‘17 -0,00022372 0,997198797 0,01342 0,98658 0
Sept. ‘17 0,000868697 0,996975704 0,01342 0,98658 0
Oct. ‘17 -0,001730693 0,997841774 0,01342 0,98658 0
Nov. ‘17 -0,001634322 0,996114816 0,01342 0,98658 0
Dec. ‘17 -0,002161012 0,994486843 0,01342 0,98658 0
Jan. ‘18 -3,90672E-05 0,992337745 0,01342 0,98658 0
Feb. ‘18 -0,000111754 0,992298977 0,01342 0,98658 0
Mar. ‘18 -0,000309253 0,992188084 0,01342 0,98658 0
April ‘18 0,001752117 0,991881247 0,01342 0,98658 0
May ‘18 -0,001120883 0,993619139 0,01342 0,98658 0
June ‘18 -0,001354472 0,992505408 0,01342 0,98658 0
July ‘18 -0,000283257 0,991161087 0,01342 0,98658 0
Aug. ‘18 -0,00176035 0,990880334 0,01342 0,98658 0
Sept. ‘18 0,000521172 0,989136038 0,01342 0,98658 0
Oct. ‘18 -0,000827312 0,989651548 0,01342 0,98658 0
Nov. ‘18 -0,001137356 0,988832798 0,01449 0,98551 0
Dec. ‘18 -0,000723619 0,987708143 0,01449 0,98551 0
Jan. ‘19 -0,00065075 0,986993419 0,01449 0,98551 0
Feb. ‘19 0,00104477 0,986351133 0,01449 0,98551 0
Mar. ‘19 0,000114 0,987381643 0,01449 0,98551 0
April ‘19 0,000441616 0,987494204 0,01449 0,98551 0
May ‘19 0,001436525 0,987930297 0,01449 0,98551 0
June ‘19 -0,000212691 0,989349484 0,01449 0,98551 0
July ‘19 0,001543428 0,989139059 0,01449 0,98551 0
Aug. ‘19 -0,00304921 0,990665724 0,01449 0,98551 0
Sept. ‘19 -0,001696144 0,987644976 0,01449 0,98551 0
Oct. ‘19 -0,000892558 0,985969788 0,01449 0,98551 0
Nov. ‘19 -0,001240872 0,985089753 0,01449 0,98551 0
Dec. ‘19 0,000417119 0,983867382 0,01449 0,98551 0
Jan. ‘20 -7,67644E-05 0,984277772 0,01449 0,98551 0
Feb. ‘20 -0,004440918 0,984202214 0,01449 0,98551 0
Mar. ‘20 0,001023008 0,979831453 0,01351 0,98649 0
April ‘20 -0,000293672 0,980833828 0,01342 0,98658 0 Table 8: Domestic portfolio - back testing
32
International Portfolio Months Portfolio
’s returns
Cumulative
portfolio’s
returns
Opt. W.
S1
Opt. W.
S2
Opt.
W.
S3
Opt.
W.
S4
Opt.
W.
S5
Opt.
W.
S6
Opt. W.
S7
Opt.
W.
S8
Opt.
W.
S9
Apr. ‘17 -0,00029 0,99970633 0,01342 0,98658 0 0 0 0 0 0 0
May ‘17 -0,00038 0,99932464 0,01342 0,98658 0 0 0 0 0 0 0
June ‘17 -0,00369 0,99563345 0,01342 0,98658 0 0 0 0 0 0 0
July ‘17 0,00096 0,99659378 0,01305 0,9868 0 0 0 0 0,00015 0 0
Aug. ‘17 0,00064 0,99723358 0,01128 0,98769 0 0 0 0 0,00103 0 0
Sep. ‘17 -0,00002 0,99721284 0,00799 0,98569 0 0 0 0 0,00632 0 0
Oct. ‘17 0,00083 0,99803731 0,01127 0,9877 0 0 0 0 0,00102 0 0
Nov. ‘17 -0,00172 0,99632185 0,00794 0,98578 0 0 0 0 0,00628 0 0
Dec. ‘17 -0,00153 0,99480088 0,01021 0,98185 0 0 0 0 0,00794 0 0
Jan. ‘18 -0,0022